Answer
$cos~(2~arctan~\frac{4}{3}) = -\frac{7}{25}$
Work Step by Step
Let $~~\theta = arctan~\frac{4}{3}$
Then: $~~tan~\theta = \frac{4}{3}$
Then: $~~sin~\theta = \frac{4}{\sqrt{3^2+4^2}} = \frac{4}{5}$
We need to find $cos~2\theta$:
$cos~2\theta = 1-2~sin^2~\theta$
$cos~2\theta = 1-2~(\frac{4}{5})^2$
$cos~2\theta = 1-\frac{32}{25}$
$cos~2\theta = -\frac{7}{25}$
Therefore, $~~cos~(2~arctan~\frac{4}{3}) = -\frac{7}{25}$
